Zeno’s paradox

Pope John Paul II "Einstein, Einstein Session of the Pontifical Academy, Vatican City (November 10, 1979). Reprinted in Science, 207, 1165-1167 (1980). Powell, C.S. Can t Get There from Here Quantum Physics Puts a New Twist on Zeno s Paradox, Sci. Amer., 24 (May 1990). 

Figure 3 Zeno s paradox. For the runner to get from point A to point B, he must cover half the distance to the point at G. To get to G, the runner must get to the halfway point at D and so on. Since there are an infinite number of halfway points, the runner can never get to B.

If you were to jump one-half the distance, then one-quarter the distance, then one-eighth the distance, and so on, will you reach the door Not in a finite number of jumps In fact, if you kept jumping forever at a rate of 1 jump per second until you are out the door, you will jump forever. Mathematically one can represent this limit of an infinite sequence of actions as the sum of the series (1/2 -i- 1/4 + 1/8 -i-...). The modem tendency is to resolve Zeno s paradox by insisting that the sum of this infinite series 1/2 -F 1/4 -F 1/8. .. is equal to 1. Because each step is done in half as much time, the actual time to complete the infinite series is no different from the real time required to leave the room. (One easy way to compute the sum of this series is to use the formula y = 1 -1"", which gives the sum of the first n terms. For n = 10, y = 0.99902.)... 

C.B. Chiu, E.C.G. Sudarshan, B. Misra, Time evolution of unstable quantum states and a resolution of Zeno s paradox, Phys. Rev. D17 (1977) 520. 

An infinite geometric series inspired by one of Zeno s paradoxes is... 

Zeno s paradox of motion claims that if you shoot an arrow, it can never reach its target. First, it has to travel half way, then half way again— meaning 1/4 of the distance—then continue with an infinite number of steps, each taking it 1/2" closer. Since infinity is so large, you will never get there. What we... 

FIGURE 2.2 The infinite series with a finite sum that is the basis of Zeno s paradox. 

Darling, David. The Universal Book of Mathematics From Abracadabra to Zeno s Paradoxes. 2004. Reprint. Edison, N.J. Castle Books, 2007. An encyclopedic account of things mathematical. 

Some facts are never stated upfront in chemistry textbooks. For example, the concept of the existence of indivisible atoms arose as a way of reconciling the paradox of having a finite entity that is composed of an infinite number of constituent elements (Zeno of Elea s paradox). Figure 2.2 shows a graphical representation of this paradox. If a frog is 1 m away from a tree and always hops half the distance between itself and the tree it will take theoretically an infinite number of hops to get there. 

Zeno paradox. On the other hand, recovering these interferences from a single path leads to excessive correlation, as evidenced by the highly oscillatory results obtained with TSH for Tully s third, extended coupling with reflection, model. This is remedied effortlessly in FMS, and one may speculate that FMS will tend to the opposite behavior Interferences that are truly present will tend to be damped if insufficient basis functions are available. This is probably preferable to the behavior seen in TSH, where there is a tendency to accentuate phase interferences and it is often unclear whether the interference effects are treated correctly. This last point can be seen in the results of the second, dual avoided crossing, model, where the TSH results exhibit oscillation, but with the wrong structure at low energies. The correct behavior can be reproduced by the FMS calculations with only ten basis functions [38]. 

The obstruction of the ion s evolution was said comprehensible, at least in principle, in terms of physical reaction of the apparatus on the ion ensemble, and as such not being too surprising. Only the non-local correlation of system and meter, and a null result of the detection, however, would exclude dynamical coupling and qualify as back-action-free measurement. Such a procedure would prove the obstruction of the evolution by measurement, that is, by gain of information, and would establish a real QZE, or "quantum Zeno paradox" (QZP) [21]. 

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